Not coplanar points in $\mathbb{R}^3$ if and only if its determinant is nonzero

Question

How to prove that points $P_1(x_1,y_1,z_1)$, $P_2(x_2,y_2,z_2)$, $P_3(x_3,y_3,z_3)$, $P_4(x_4,y_4,z_4)$ are not coplanar if and only if the determinant $$\left|\begin{array}{cccc} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 &1 \\ \end{array} \right| \neq 0$$

Answer 1

If the points are coplanar, then you can find the equation of a plane $P \equiv ax+by+cz+d=0$ such that your points all verify the equation with $(a,b,c,d) \neq (0,0,0,0)$. Hence the system $$\begin{cases} x_1 a + y_1 b + z_1 c + 1 d = 0 \\ x_2 a + y_2 b + z_2 c + 1 d = 0 \\ x_3 a + y_3 b + z_3 c + 1 d = 0 \\ x_4 a + y_4 b + z_4 c + 1 d = 0 \end{cases}$$ has a non trivial solution and the determinant vanishes (if not $(0,0,0,0)$ would be the only solution).

Conversely, if the determinant vanishes, the system has at least a non zero solution $(a,b,c,d)$ and this forms the equation of a plane satisfied by the points which are coplanar.

Answer 2

Claim:

The absolute value of this determinant
$$\left|\begin{array}{cccc} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 &1 \\ \end{array} \right| $$
is the volume of a (skew) parallelepiped with vertices $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3),(x_4,y_4,z_4)$.

Proof: We know that the volume is the absolute value of the determinat that has as columns the vectors that represents the sides of the parallelepiped: $$ d= \left|\begin{array}{cccc} x_1-x_4 & y_1-y_4 & z_1-z_4 \\ x_2-x_4 & y_2-y_4 & z_2-z_4 \\ x_3-x_4 & y_3-y_4 & z_3-z_4\\ \end{array} \right| $$ Now, by co-factor decomposition of a determinant we have: $$ d= \left|\begin{array}{cccc} x_1-x_4 & y_1-y_4 & z_1-z_4&1 \\ x_2-x_4 & y_2-y_4 & z_2-z_4&1 \\ x_3-x_4 & y_3-y_4 & z_3-z_4&1\\ 0&0&0&1 \end{array} \right| $$ and, multiply the last columns bi $x_4$ and adding to the frst column we find $$ d= \left|\begin{array}{cccc} x_1 & y_1-y_4 & z_1-z_4&1 \\ x_2 & y_2-y_4 & z_2-z_4&1 \\ x_3 & y_3-y_4 & z_3-z_4&1\\ x_4&0&0&1 \end{array} \right| $$ Doing the same,i.e. multiplying by $y_4$ and $z_4$ and adding to the second and third columns we arrive at: $$ d= \left|\begin{array}{cccc} x_1 & y_1 & z_1&1 \\ x_2 & y_2 & z_2&1 \\ x_3 & y_3 & z_3&1\\ x_4&y_4&z_4&1 \end{array} \right| $$

That prove the claim.

Now the answer is obvious since the volume is null iff the four points are coplanar.